Question: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{-2r^3 + 30r^2 - 112r}{-7r^3 + 70r^2 - 147r}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {-2r(r^2 - 15r + 56)} {-7r(r^2 - 10r + 21)} $ $ x = \dfrac{2r}{7r} \cdot \dfrac{r^2 - 15r + 56}{r^2 - 10r + 21} $ Simplify: $ x = \dfrac{2}{7} \cdot \dfrac{r^2 - 15r + 56}{r^2 - 10r + 21}$ Since we are dividing by $r$ , we must remember that $r \neq 0$ Next factor the numerator and denominator. $ x = \dfrac{2}{7} \cdot \dfrac{(r - 7)(r - 8)}{(r - 7)(r - 3)}$ Assuming $r \neq 7$ , we can cancel the $r - 7$ $ x = \dfrac{2}{7} \cdot \dfrac{r - 8}{r - 3}$ Therefore: $ x = \dfrac{ 2(r - 8)}{ 7(r - 3)}$, $r \neq 7$, $r \neq 0$